| L(s) = 1 | + 3·2-s − 18·3-s + 4·4-s − 33·5-s − 54·6-s − 183·8-s + 81·9-s − 99·10-s + 1.13e3·11-s − 72·12-s − 1.85e3·13-s + 594·15-s + 187·16-s − 324·17-s + 243·18-s + 2.31e3·19-s − 132·20-s + 3.41e3·22-s − 1.59e3·23-s + 3.29e3·24-s + 3.47e3·25-s − 5.55e3·26-s + 1.45e3·27-s − 4.43e3·29-s + 1.78e3·30-s + 4.29e3·31-s − 1.11e3·32-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 1.15·3-s + 1/8·4-s − 0.590·5-s − 0.612·6-s − 1.01·8-s + 1/3·9-s − 0.313·10-s + 2.83·11-s − 0.144·12-s − 3.03·13-s + 0.681·15-s + 0.182·16-s − 0.271·17-s + 0.176·18-s + 1.46·19-s − 0.0737·20-s + 1.50·22-s − 0.629·23-s + 1.16·24-s + 1.11·25-s − 1.61·26-s + 0.384·27-s − 0.979·29-s + 0.361·30-s + 0.802·31-s − 0.192·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.310430697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.310430697\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 T + 5 T^{2} + 45 p^{2} T^{3} - 81 p^{4} T^{4} + 45 p^{7} T^{5} + 5 p^{10} T^{6} - 3 p^{15} T^{7} + p^{20} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 33 T - 2383 T^{2} - 91674 T^{3} - 1113966 T^{4} - 91674 p^{5} T^{5} - 2383 p^{10} T^{6} + 33 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1137 T + 58867 p T^{2} - 367398810 T^{3} + 182185169004 T^{4} - 367398810 p^{5} T^{5} + 58867 p^{11} T^{6} - 1137 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 925 T + 951450 T^{2} + 925 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 324 T - 1397458 T^{2} - 433278720 T^{3} + 92263115475 T^{4} - 433278720 p^{5} T^{5} - 1397458 p^{10} T^{6} + 324 p^{15} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2311 T - 901277 T^{2} - 2980727800 T^{3} + 19607318846824 T^{4} - 2980727800 p^{5} T^{5} - 901277 p^{10} T^{6} - 2311 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 1596 T - 5056990 T^{2} - 8408494080 T^{3} + 2681805699 T^{4} - 8408494080 p^{5} T^{5} - 5056990 p^{10} T^{6} + 1596 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2217 T + 42125014 T^{2} + 2217 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4294 T - 6712931 T^{2} + 137867178890 T^{3} - 714913793459948 T^{4} + 137867178890 p^{5} T^{5} - 6712931 p^{10} T^{6} - 4294 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 19109 T + 180683803 T^{2} - 874851371876 T^{3} + 3899410282461118 T^{4} - 874851371876 p^{5} T^{5} + 180683803 p^{10} T^{6} - 19109 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12858 T + 228491122 T^{2} - 12858 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 2771 T + 36114396 T^{2} + 2771 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 23160 T - 33691018 T^{2} + 2579713748640 T^{3} + 149370858967788675 T^{4} + 2579713748640 p^{5} T^{5} - 33691018 p^{10} T^{6} + 23160 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 31653 T + 105901883 T^{2} - 1887137299620 T^{3} + 208952138512710990 T^{4} - 1887137299620 p^{5} T^{5} + 105901883 p^{10} T^{6} - 31653 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 41097 T + 791968799 T^{2} + 21898700344836 T^{3} - 913998404147564592 T^{4} + 21898700344836 p^{5} T^{5} + 791968799 p^{10} T^{6} - 41097 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 42052 T + 39963442 T^{2} - 1649054882320 T^{3} + 780489396276288139 T^{4} - 1649054882320 p^{5} T^{5} + 39963442 p^{10} T^{6} - 42052 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 30763 T - 751671155 T^{2} - 30831198187070 T^{3} - 217227959819149556 T^{4} - 30831198187070 p^{5} T^{5} - 751671155 p^{10} T^{6} + 30763 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 102096 T + 6091648810 T^{2} - 102096 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 28577 T - 1819854179 T^{2} - 43141098817006 T^{3} + 960538312903836118 T^{4} - 43141098817006 p^{5} T^{5} - 1819854179 p^{10} T^{6} + 28577 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18464 T - 2991126965 T^{2} + 52106636539168 T^{3} + 585228301517857816 T^{4} + 52106636539168 p^{5} T^{5} - 2991126965 p^{10} T^{6} - 18464 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 61179 T + 8589312784 T^{2} + 61179 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 29322 T - 10522232110 T^{2} - 6271767496512 T^{3} + 93567411680290480719 T^{4} - 6271767496512 p^{5} T^{5} - 10522232110 p^{10} T^{6} - 29322 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 9791 T + 17134261944 T^{2} - 9791 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.584627435761080225252813523216, −8.411963297128130377393152855156, −7.917992529527696500683220263382, −7.63128611379144953824655800406, −7.53952866227194424061095752831, −7.05581357304159740614446503731, −6.71067307135083434897238490375, −6.58351155149557264993735963525, −6.53119287453275139723485155541, −5.83627597816343607064250837927, −5.68020654222288376945633244489, −5.45644156454027251844159661587, −5.09342162660238438750609300709, −4.61884884009651965233427604431, −4.61171376266579658724462522695, −3.87018363236734616704635388603, −3.81036429935956378440335847702, −3.62432673740447448215357559474, −2.73618795758857292681513818733, −2.63416789409053981479992505084, −2.35345011498053300397345305235, −1.48116380786680013415431719406, −1.06450552720894624047844550173, −0.57465756086427932340667051695, −0.45387600516148004167832666732,
0.45387600516148004167832666732, 0.57465756086427932340667051695, 1.06450552720894624047844550173, 1.48116380786680013415431719406, 2.35345011498053300397345305235, 2.63416789409053981479992505084, 2.73618795758857292681513818733, 3.62432673740447448215357559474, 3.81036429935956378440335847702, 3.87018363236734616704635388603, 4.61171376266579658724462522695, 4.61884884009651965233427604431, 5.09342162660238438750609300709, 5.45644156454027251844159661587, 5.68020654222288376945633244489, 5.83627597816343607064250837927, 6.53119287453275139723485155541, 6.58351155149557264993735963525, 6.71067307135083434897238490375, 7.05581357304159740614446503731, 7.53952866227194424061095752831, 7.63128611379144953824655800406, 7.917992529527696500683220263382, 8.411963297128130377393152855156, 8.584627435761080225252813523216
